Advertisement

Image Reconstruction by Multilabel Propagation

  • Matthias ZislerEmail author
  • Freddie Åström
  • Stefania Petra
  • Christoph Schnörr
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10302)

Abstract

This work presents a non-convex variational approach to joint image reconstruction and labeling. Our regularization strategy, based on the KL-divergence, takes into account the smooth geometry on the space of discrete probability distributions. The proposed objective function is efficiently minimized via DC programming which amounts to solving a sequence of convex programs, with guaranteed convergence to a critical point. Each convex program is solved by a generalized primal dual algorithm. This entails the evaluation of a proximal mapping, evaluated efficiently by a fixed point iteration. We illustrate our approach on few key scenarios in discrete tomography and image deblurring.

Keywords

Convex Relaxation Proximal Mapping Discrete Probability Distribution Fixed Point Iteration Discrete Tomography 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Hanke, R., Fuchs, T., Uhlmann, N.: X-ray based methods for non-destructive testing and material characterization. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 591(1), 14–18 (2008)CrossRefGoogle Scholar
  2. 2.
    Zach, C., Gallup, D., Frahm, J., Niethammer, M.: Fast global labeling for real-time stereo using multiple plane sweeps. In: VMV, pp. 243–252 (2008)Google Scholar
  3. 3.
    Chambolle, A., Cremers, D., Pock, T.: A convex approach to minimal partitions. SIAM J. Imaging Sci. 5(4), 1113–1158 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Lellmann, J., Kappes, J., Yuan, J., Becker, F., Schnörr, C.: Convex multi-class image labeling by simplex-constrained total variation. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 150–162. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-02256-2_13 CrossRefGoogle Scholar
  5. 5.
    Lellmann, J., Schnörr, C.: Continuous multiclass labeling approaches and algorithms. SIAM J. Imaging Sci. 4(4), 1049–1096 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Åström, F., Petra, S., Schmitzer, B., Schnörr, C.: Image labeling by assignment. J. Math. Imaging Vis. 58, 1–28 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bergmann, R., Fitschen, J.H., Persch, J., Steidl, G.: Iterative multiplicative filters for data labeling. Int. J. Comput. Vis., 1–19 (2017)Google Scholar
  8. 8.
    Martin, D., Fowlkes, C., Tal, D., Malik, J.: A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics. In: Proceedings of the ICCV, pp. 416–423 (2001)Google Scholar
  9. 9.
    Zisler, M., Kappes, J.H., Schnörr, C., Petra, S., Schnörr, C.: Non-binary discrete tomography by continuous non-convex optimization. IEEE Trans. Comput. Imaging 2(3), 335–347 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Zisler, M., Petra, S., Schnörr, C., Schnörr, C.: Discrete tomography by continuous multilabeling subject to projection constraints. In: Rosenhahn, B., Andres, B. (eds.) GCPR 2016. LNCS, vol. 9796, pp. 261–272. Springer, Cham (2016). doi: 10.1007/978-3-319-45886-1_21 CrossRefGoogle Scholar
  11. 11.
    Kass, R.E.: The geometry of asymptotic inference. Stat. Sci. 4(3), 188–234 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Amari, S.I., Cichocki, A.: Information geometry of divergence functions. Bull. Pol. Acad. Sci.: Tech. Sci. 58(1), 183–195 (2010)Google Scholar
  13. 13.
    Potts, R.B.: Some generalized order-disorder transformations. Math. Proc. Camb. Philos. Soc. 48, 106–109 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Weinmann, A., Demaret, L., Storath, M.: Total variation regularization for manifold-valued data. SIAM J. Imaging Sci. 7(4), 2226–2257 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cover, T., Thomas, J.: Elements of Information Theory, 2nd edn. Wiley, Hoboken (2006)zbMATHGoogle Scholar
  16. 16.
    Pham Dinh, T., El Bernoussi, S.: Algorithms for solving a class of nonconvex optimization problems. Methods of subgradients. In: Hiriart-Urruty, J.B. (ed.) Fermat Days 85: Mathematics for Optimization. North-Holland Mathematics Studies, vol. 129, pp. 249–271. North-Holland, Amsterdam (1986)CrossRefGoogle Scholar
  17. 17.
    Pham-Dinh, T., Hoai An, L.: Convex analysis approach to D.C. programming: theory, algorithms and applications. Acta Math. Vietnam. 22(1), 289–355 (1997)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Heidelberg (2009)zbMATHGoogle Scholar
  19. 19.
    Chambolle, A., Pock, T.: On the ergodic convergence rates of a first-order primal-dual algorithm. Math. Program. 159(1), 253–287 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Batenburg, K., Sijbers, J.: DART: a practical reconstruction algorithm for discrete tomography. IEEE Trans. Image Process. 20(9), 2542–2553 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Varga, L., Balázs, P., Nagy, A.: An energy minimization reconstruction algorithm for multivalued discrete tomography. In: 3rd International Symposium on Computational Modeling of Objects Represented in Images, Italy, pp. 179–185 (2012)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Matthias Zisler
    • 1
    Email author
  • Freddie Åström
    • 1
  • Stefania Petra
    • 2
  • Christoph Schnörr
    • 1
  1. 1.Image and Pattern Analysis GroupHeidelberg UniversityHeidelbergGermany
  2. 2.Mathematical Imaging GroupHeidelberg UniversityHeidelbergGermany

Personalised recommendations